src/HOL/Quickcheck_Random.thy
 author paulson Tue Apr 25 16:39:54 2017 +0100 (2017-04-25) changeset 65578 e4997c181cce parent 62979 1e527c40ae40 child 67399 eab6ce8368fa permissions -rw-r--r--
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
```     1 (*  Title:      HOL/Quickcheck_Random.thy
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```     2     Author:     Florian Haftmann & Lukas Bulwahn, TU Muenchen
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```     3 *)
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```     4
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```     5 section \<open>A simple counterexample generator performing random testing\<close>
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```     6
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```     7 theory Quickcheck_Random
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```     8 imports Random Code_Evaluation Enum
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```     9 begin
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```    10
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```    11 notation fcomp (infixl "\<circ>>" 60)
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```    12 notation scomp (infixl "\<circ>\<rightarrow>" 60)
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```    13
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```    14 setup \<open>Code_Target.add_derived_target ("Quickcheck", [(Code_Runtime.target, I)])\<close>
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```    15
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```    16 subsection \<open>Catching Match exceptions\<close>
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```    17
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```    18 axiomatization catch_match :: "'a => 'a => 'a"
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```    19
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```    20 code_printing
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```    21   constant catch_match \<rightharpoonup> (Quickcheck) "((_) handle Match => _)"
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```    22
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```    23 subsection \<open>The \<open>random\<close> class\<close>
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```    24
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```    25 class random = typerep +
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```    26   fixes random :: "natural \<Rightarrow> Random.seed \<Rightarrow> ('a \<times> (unit \<Rightarrow> term)) \<times> Random.seed"
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```    27
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```    28
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```    29 subsection \<open>Fundamental and numeric types\<close>
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```    30
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```    31 instantiation bool :: random
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```    32 begin
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```    33
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```    34 definition
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```    35   "random i = Random.range 2 \<circ>\<rightarrow>
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```    36     (\<lambda>k. Pair (if k = 0 then Code_Evaluation.valtermify False else Code_Evaluation.valtermify True))"
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```    37
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```    38 instance ..
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```    39
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```    40 end
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```    41
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```    42 instantiation itself :: (typerep) random
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```    43 begin
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```    44
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```    45 definition
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```    46   random_itself :: "natural \<Rightarrow> Random.seed \<Rightarrow> ('a itself \<times> (unit \<Rightarrow> term)) \<times> Random.seed"
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```    47 where "random_itself _ = Pair (Code_Evaluation.valtermify TYPE('a))"
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```    48
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```    49 instance ..
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```    50
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```    51 end
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```    52
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```    53 instantiation char :: random
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```    54 begin
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```    55
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```    56 definition
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```    57   "random _ = Random.select (Enum.enum :: char list) \<circ>\<rightarrow> (\<lambda>c. Pair (c, \<lambda>u. Code_Evaluation.term_of c))"
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```    58
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```    59 instance ..
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```    60
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```    61 end
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```    62
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```    63 instantiation String.literal :: random
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```    64 begin
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```    65
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```    66 definition
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```    67   "random _ = Pair (STR '''', \<lambda>u. Code_Evaluation.term_of (STR ''''))"
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```    68
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```    69 instance ..
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```    70
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```    71 end
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```    72
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```    73 instantiation nat :: random
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```    74 begin
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```    75
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```    76 definition random_nat :: "natural \<Rightarrow> Random.seed
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```    77   \<Rightarrow> (nat \<times> (unit \<Rightarrow> Code_Evaluation.term)) \<times> Random.seed"
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```    78 where
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```    79   "random_nat i = Random.range (i + 1) \<circ>\<rightarrow> (\<lambda>k. Pair (
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```    80      let n = nat_of_natural k
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```    81      in (n, \<lambda>_. Code_Evaluation.term_of n)))"
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```    82
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```    83 instance ..
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```    84
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```    85 end
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```    86
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```    87 instantiation int :: random
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```    88 begin
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```    89
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```    90 definition
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```    91   "random i = Random.range (2 * i + 1) \<circ>\<rightarrow> (\<lambda>k. Pair (
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```    92      let j = (if k \<ge> i then int (nat_of_natural (k - i)) else - (int (nat_of_natural (i - k))))
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```    93      in (j, \<lambda>_. Code_Evaluation.term_of j)))"
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```    94
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```    95 instance ..
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```    96
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```    97 end
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```    98
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```    99 instantiation natural :: random
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```   100 begin
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```   101
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```   102 definition random_natural :: "natural \<Rightarrow> Random.seed
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```   103   \<Rightarrow> (natural \<times> (unit \<Rightarrow> Code_Evaluation.term)) \<times> Random.seed"
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```   104 where
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```   105   "random_natural i = Random.range (i + 1) \<circ>\<rightarrow> (\<lambda>n. Pair (n, \<lambda>_. Code_Evaluation.term_of n))"
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```   106
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```   107 instance ..
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```   108
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```   109 end
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```   110
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```   111 instantiation integer :: random
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```   112 begin
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```   113
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```   114 definition random_integer :: "natural \<Rightarrow> Random.seed
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```   115   \<Rightarrow> (integer \<times> (unit \<Rightarrow> Code_Evaluation.term)) \<times> Random.seed"
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```   116 where
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```   117   "random_integer i = Random.range (2 * i + 1) \<circ>\<rightarrow> (\<lambda>k. Pair (
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```   118      let j = (if k \<ge> i then integer_of_natural (k - i) else - (integer_of_natural (i - k)))
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```   119       in (j, \<lambda>_. Code_Evaluation.term_of j)))"
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```   120
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```   121 instance ..
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```   122
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```   123 end
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```   124
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```   125
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```   126 subsection \<open>Complex generators\<close>
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```   127
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```   128 text \<open>Towards @{typ "'a \<Rightarrow> 'b"}\<close>
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```   129
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```   130 axiomatization random_fun_aux :: "typerep \<Rightarrow> typerep \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> term)
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```   131   \<Rightarrow> (Random.seed \<Rightarrow> ('b \<times> (unit \<Rightarrow> term)) \<times> Random.seed)
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```   132   \<Rightarrow> (Random.seed \<Rightarrow> Random.seed \<times> Random.seed)
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```   133   \<Rightarrow> Random.seed \<Rightarrow> (('a \<Rightarrow> 'b) \<times> (unit \<Rightarrow> term)) \<times> Random.seed"
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```   134
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```   135 definition random_fun_lift :: "(Random.seed \<Rightarrow> ('b \<times> (unit \<Rightarrow> term)) \<times> Random.seed)
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```   136   \<Rightarrow> Random.seed \<Rightarrow> (('a::term_of \<Rightarrow> 'b::typerep) \<times> (unit \<Rightarrow> term)) \<times> Random.seed"
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```   137 where
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```   138   "random_fun_lift f =
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```   139     random_fun_aux TYPEREP('a) TYPEREP('b) (op =) Code_Evaluation.term_of f Random.split_seed"
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```   140
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```   141 instantiation "fun" :: ("{equal, term_of}", random) random
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```   142 begin
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```   143
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```   144 definition
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```   145   random_fun :: "natural \<Rightarrow> Random.seed \<Rightarrow> (('a \<Rightarrow> 'b) \<times> (unit \<Rightarrow> term)) \<times> Random.seed"
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```   146   where "random i = random_fun_lift (random i)"
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```   147
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```   148 instance ..
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```   149
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```   150 end
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```   151
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```   152 text \<open>Towards type copies and datatypes\<close>
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```   153
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```   154 definition collapse :: "('a \<Rightarrow> ('a \<Rightarrow> 'b \<times> 'a) \<times> 'a) \<Rightarrow> 'a \<Rightarrow> 'b \<times> 'a"
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```   155   where "collapse f = (f \<circ>\<rightarrow> id)"
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```   156
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```   157 definition beyond :: "natural \<Rightarrow> natural \<Rightarrow> natural"
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```   158   where "beyond k l = (if l > k then l else 0)"
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```   159
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```   160 lemma beyond_zero: "beyond k 0 = 0"
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```   161   by (simp add: beyond_def)
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```   162
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```   163
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```   164 definition (in term_syntax) [code_unfold]:
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```   165   "valterm_emptyset = Code_Evaluation.valtermify ({} :: ('a :: typerep) set)"
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```   166
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```   167 definition (in term_syntax) [code_unfold]:
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```   168   "valtermify_insert x s = Code_Evaluation.valtermify insert {\<cdot>} (x :: ('a :: typerep * _)) {\<cdot>} s"
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```   169
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```   170 instantiation set :: (random) random
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```   171 begin
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```   172
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```   173 fun random_aux_set
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```   174 where
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```   175   "random_aux_set 0 j = collapse (Random.select_weight [(1, Pair valterm_emptyset)])"
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```   176 | "random_aux_set (Code_Numeral.Suc i) j =
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```   177     collapse (Random.select_weight
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```   178       [(1, Pair valterm_emptyset),
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```   179        (Code_Numeral.Suc i,
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```   180         random j \<circ>\<rightarrow> (%x. random_aux_set i j \<circ>\<rightarrow> (%s. Pair (valtermify_insert x s))))])"
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```   181
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```   182 lemma [code]:
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```   183   "random_aux_set i j =
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```   184     collapse (Random.select_weight [(1, Pair valterm_emptyset),
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```   185       (i, random j \<circ>\<rightarrow> (%x. random_aux_set (i - 1) j \<circ>\<rightarrow> (%s. Pair (valtermify_insert x s))))])"
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```   186 proof (induct i rule: natural.induct)
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```   187   case zero
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```   188   show ?case by (subst select_weight_drop_zero [symmetric])
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```   189     (simp add: random_aux_set.simps [simplified] less_natural_def)
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```   190 next
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```   191   case (Suc i)
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```   192   show ?case by (simp only: random_aux_set.simps(2) [of "i"] Suc_natural_minus_one)
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```   193 qed
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```   194
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```   195 definition "random_set i = random_aux_set i i"
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```   196
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```   197 instance ..
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```   198
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```   199 end
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```   200
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```   201 lemma random_aux_rec:
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```   202   fixes random_aux :: "natural \<Rightarrow> 'a"
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```   203   assumes "random_aux 0 = rhs 0"
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```   204     and "\<And>k. random_aux (Code_Numeral.Suc k) = rhs (Code_Numeral.Suc k)"
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```   205   shows "random_aux k = rhs k"
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```   206   using assms by (rule natural.induct)
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```   207
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```   208 subsection \<open>Deriving random generators for datatypes\<close>
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```   209
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```   210 ML_file "Tools/Quickcheck/quickcheck_common.ML"
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```   211 ML_file "Tools/Quickcheck/random_generators.ML"
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```   212
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```   213
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```   214 subsection \<open>Code setup\<close>
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```   215
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```   216 code_printing
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```   217   constant random_fun_aux \<rightharpoonup> (Quickcheck) "Random'_Generators.random'_fun"
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```   218   \<comment> \<open>With enough criminal energy this can be abused to derive @{prop False};
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```   219   for this reason we use a distinguished target \<open>Quickcheck\<close>
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```   220   not spoiling the regular trusted code generation\<close>
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```   221
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```   222 code_reserved Quickcheck Random_Generators
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```   223
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```   224 no_notation fcomp (infixl "\<circ>>" 60)
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```   225 no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
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```   226
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```   227 hide_const (open) catch_match random collapse beyond random_fun_aux random_fun_lift
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```   228
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```   229 hide_fact (open) collapse_def beyond_def random_fun_lift_def
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```   230
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```   231 end
```